IMathAS-libretexts

Consider the indefinite integral

\displaystyle \int{\frac{{{3}{x}^{{3}}+{4}{x}^{{2}}-{6}{x}-{12}}}{{{x}^{{2}}-{4}}}},{\left.{d}{x}\right.}

Then the integrand decomposes into the form

\displaystyle {a}{x}+{b}+{\frac{{{c}}}{{{x}-{2}}}}+{\frac{{{d}}}{{{x}+{2}}}}

where

\displaystyle {a}

\displaystyle {b}

\displaystyle {c}

\displaystyle {d}

=
Integrating term by term, we obtain that

\displaystyle \int{\frac{{{3}{x}^{{3}}+{4}{x}^{{2}}-{6}{x}-{12}}}{{{x}^{{2}}-{4}}}},{\left.{d}{x}\right.}=

\displaystyle +{C}